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The aspects of gravity modelling with a semi-infinite vertical line of mass as an approximation to the model of semi-infinite vertical cylinder is critically reviewed. The brief outline of the derivations of gravity response due to both type of models, such as the semi-infinite vertical line of mass and vertical right circular cylinder, is given. The model of semi-infinite vertical right circular cylinder provides no closed form formula to compute gravity response on a free horizontal surface, and the ensuing formulation is comprised of elliptic integrals of the first and second kinds. On the other hand, there exists closed form formula of computing gravity response due to semi-infinite vertical line of mass that approximates closely to the semi-infinite vertical thin rod. Assessment on the limitation of replacing vertical cylinder model by vertical line of mass both by rigorous theoretical discourse and numerical tests is made. It is shown via numerical experiment that if the radius of cross section of the vertical right circular cylinder is one-tenth of its depth of burial, then the approximation of line of mass model would provide reasonably accurate results in the order of 10 - 4 . [ABSTRACT FROM AUTHOR]

An aerodynamic problem on an air flow around a large aspect ratio rectangular wing is investigated in this study. According to the theory of Vlasov, the wing is considered to be a thin rod. External loads are assumed to be proportional to the airfoil angle of attack related to the dimensionless coefficient of the lift and the pitching moment coefficient. These coefficients depend on the airfoil parameters and the Mach number M and are determined by experimental measurements for subsonic and supersonic velocities. In this case, to define the unstable cases of the wing, one bases on the Lyapunov stability theory. Equations of bending and torsional free vibrations have resulted. Based on the analysis of natural frequencies (eigenfrequencies), it is possible to determine the changing positions of the real part and the imaginary part of the characteristic equation solution. These positions can cause instabilities for the wing such as torsional divergence and flutter.

For the sake of intended industrial applications, the stress wave propagation in thin rods of brittle material is measured and explored. Based on the known theory of a non-dispersive wave propagation in rods, a numerically efficient method for the calculation of wave propagation and reflection is presented. The effects of an impact applied to a thin rod are tested and measured on a test bench. The results of these measurements carried out on glass capillary tubes are successfully compared with theoretical results and they confirm the possibility of brittle rod cutting by means of elastic waves with the required fracture surface quality.

In this paper, a mechanical system composed of a thin rotating rod under some constraints is considered. For this system, the total torque of the gravity forces is fixed and the unknown function to be determined is the mass density of the rod. This kind of problem is faced in several engineering applications as in aerospace. The resulting problem is formulated as a non classical integral equation, where the conventional methods of resolution do not apply. Therefore, a special treatment is required to solve the obtained integral equation. First, the obtained integral equation is transformed into a system of mixed integral and linear differential equations with two unknown functions. The latter transformation allows the inspiration of the general expression of the requested functions. Consequently, a highly non linear system with several unknowns is obtained. During the resolution of the latter system several mathematical technics are used. After applying all these technics an analytical solution of the studied integral equation is obtained. Finally, the technical feasibility from an engineering viewpoint of the production of a thin rod with the obtained mass density function is briefly discussed. In this context, the Functionally Graded Material is proposed as a material satisfying the obtained mass density function.

We derive a one‐dimensional model for an elastic shuttle, that is, a thin rod with rounded ends and small fixed terminals, by means of an asymptotic procedure of dimension reduction. In the model, deformation of the shuttle is described by a system of ordinary differential equations with variable degenerating coefficients, and the number of the required boundary conditions at the end points of the one‐dimensional image of the rod depends on the roundness exponent m∈(0,1). Error estimates are obtained in the case m∈(0,1/4) by using an anisotropic weighted Korn inequality, which was derived in an earlier paper by the authors. We also briefly discuss boundary layer effects, which can be neglected in the case m∈(0,1/4) but play a crucial role in the formulation of the limit problem for m ≥ 1/4. [ABSTRACT FROM AUTHOR]

The dynamics of a thin rod under the action of compressive force is considered. The compressive force increases abruptly and then remains constant. The compression of the rod is assumed to take place instantly throughout its length. The motion of the rod is studied, depending on the magnitude and time of action of the compressive force defined by the phase of the initial free oscillation of the rod. Only the initial stage of the process is investigated, for which the linear bending theory is valid. The friction forces are not taken into account. An essential relationship is shown to exist between the dynamics of the rod and the initial conditions determined by the phase of the bending oscillations at the instant of impact. Relatively weak impact leads to excitation of the bending oscillations. Rearrangement of harmonics develops with time under specified initial maximum deflection from a straight line and zero velocity. The fastest buckling takes place at some initial deflection with the velocity directed toward the increasing bend. [ABSTRACT FROM AUTHOR]

We consider the barotropic Navier-Stokes system describing the motion of a compressible viscous fluid confined to a cavity shaped as a thin rod Ω= εQ×(0,1), Q⊂ R. We show that the weak solutions in the 3D domain converge to (strong) solutions of the limit 1D Navier-Stokes system as ε→0. [ABSTRACT FROM AUTHOR]

A nondestructive method is proposed to measure the elastic constants of thin rods. The determined Young's modulus increases linearly with the increment in the chemical content (Al 2 O 3 , MgO, TFe 2 O 3 , and TiO 2) of the rods. The accuracy of the proposed method is 3 GPa, while the relative error of the proposed method is less than 4.30%. [Display omitted] Thin rods have been widely used in mechanical systems. The elastic constants, such as Young's modulus and Poisson's ratio, are of great interest for evaluating the quality of thin rods. This paper proposes a nondestructive method, utilizing longitudinal guided waves at low frequency and flexural guided waves at high frequency, to approximate bar waves and transverse bulk waves, respectively. According to the approximate velocities of bar waves and transverse bulk waves, the elastic constants can be determined inversely. Both simulation and experiment are conducted on thin aluminum rods with known elastic constants. The measurement errors are less than 2.03%, demonstrating the correctness of proposed method. Then, the proposed method is applied on 20 thin basalt rods and 30 basalt fibers. The incident frequency ranges from 0.3 to 1.8 MHz, and the diameter of thin rods ranges from 0.58 to 1.51 mm. The determined Young's modulus is 4.30% higher than the value from the destructive tensile tests, and the determined Young's moduli increase linearly with the chemical content (Al 2 O 3 , MgO, TFe 2 O 3 , and TiO 2). The proposed method and associated experimental and simulation techniques can be used in the quality control and nondestructive testing of rod-like structures made by homogenous isotropic linear elastic materials. [ABSTRACT FROM AUTHOR]

Abstract: The paper deals with the isogeometric analysis via B-splines of space rods under Kirchhoff-Love hypotheses. The approach was used by Gontier and Vollmer [12] for developing a plane curve element within the framework of the Timoshenko rod model, but they adopted only one patch to represent entirely the geometry of the rod; furthermore the authors developed their theory only for plane elements. In this work we develop an isogeometric approach for the numerical analysis of the 3D Kirchhoff-Love rod theory. We use B-splines and Bezier interpolations and we show that they are able to attain very good accuracy for rod structures, particularly for developing a 3D exact curve element with geometric torsion. The paper presents an original parametrization of the geometric torsion that proves to be very effective. The use of B-splines allows to avoid discontinuities on the geometrical quantities, and particularly on the normal fields, so that even relatively low order interpolation functions are able to yield accurate results. [Copyright &y& Elsevier]

We study the asymptotic behaviour of the displacement of a thin periodically perforated rod under the action of forces applied to one of the rod ends, another end of the rod is clamped. We show that, up to boundary layer functions arising in the vicinity of the end points of the rod, the set of solutions forms a finite dimensional space, and that in the interior of the rod any solution only depends on the resultant forces and moments. We also provide the results of numerical computations of the effective torsion rigidity for a hexagonal periodically perforated rod. [ABSTRACT FROM AUTHOR]

In this paper we apply the asymptotic expansion method to obtain a nonlinear adaptive elastic rod model. We first consider the model of Cowin and Hegedus with later modifications, and with a remodeling rate equation depending nonlinearly on the strain field and for a thin rod whose cross section is a function of a small parameter. Based on the asymptotic expansion method for the elastic case, we prove that, when the small parameter tends to zero the solution of the nonlinear adaptive elastic rod model converges to the leading term of its asymptotic expansion. Moreover, we show that this term is also the solution of a well-known simplified adaptive elastic model, with generalized Bernoulli-Navier equilibrium equations and a remodeling rate equation whose driving mechanism is the strain energy per unit volume, in good agreement with some of the models used in practice. [ABSTRACT FROM AUTHOR]

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